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annotate src/video/e_sqrt.h @ 1643:51038e80ae59

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More general fix for bug #189 The clipping is done at a higher level, and the low level functions are passed clipped rectangles. Drivers which don't support source clipping have not been changed, so the image will be squished instead of clipped, but at least they will no longer crash when the destination rect was out of bounds.
author Sam Lantinga <slouken@libsdl.org>
date Mon, 17 Apr 2006 06:47:23 +0000
parents 7a610f25c12f
children 782fd950bd46 c121d94672cb
rev   line source
1330
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1 /* @(#)e_sqrt.c 5.1 93/09/24 */
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2 /*
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3 * ====================================================
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4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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5 *
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6 * Developed at SunPro, a Sun Microsystems, Inc. business.
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7 * Permission to use, copy, modify, and distribute this
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8 * software is freely granted, provided that this notice
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9 * is preserved.
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10 * ====================================================
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11 */
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12
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13 #if defined(LIBM_SCCS) && !defined(lint)
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14 static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
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15 #endif
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16
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17 /* __ieee754_sqrt(x)
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18 * Return correctly rounded sqrt.
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19 * ------------------------------------------
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20 * | Use the hardware sqrt if you have one |
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21 * ------------------------------------------
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22 * Method:
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23 * Bit by bit method using integer arithmetic. (Slow, but portable)
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24 * 1. Normalization
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25 * Scale x to y in [1,4) with even powers of 2:
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26 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
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27 * sqrt(x) = 2^k * sqrt(y)
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28 * 2. Bit by bit computation
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29 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
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30 * i 0
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31 * i+1 2
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32 * s = 2*q , and y = 2 * ( y - q ). (1)
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33 * i i i i
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34 *
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35 * To compute q from q , one checks whether
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36 * i+1 i
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37 *
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38 * -(i+1) 2
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39 * (q + 2 ) <= y. (2)
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40 * i
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41 * -(i+1)
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42 * If (2) is false, then q = q ; otherwise q = q + 2 .
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43 * i+1 i i+1 i
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44 *
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45 * With some algebric manipulation, it is not difficult to see
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46 * that (2) is equivalent to
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47 * -(i+1)
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48 * s + 2 <= y (3)
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49 * i i
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50 *
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51 * The advantage of (3) is that s and y can be computed by
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52 * i i
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53 * the following recurrence formula:
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54 * if (3) is false
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55 *
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56 * s = s , y = y ; (4)
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57 * i+1 i i+1 i
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58 *
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59 * otherwise,
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60 * -i -(i+1)
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61 * s = s + 2 , y = y - s - 2 (5)
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62 * i+1 i i+1 i i
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63 *
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64 * One may easily use induction to prove (4) and (5).
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65 * Note. Since the left hand side of (3) contain only i+2 bits,
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66 * it does not necessary to do a full (53-bit) comparison
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67 * in (3).
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68 * 3. Final rounding
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69 * After generating the 53 bits result, we compute one more bit.
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70 * Together with the remainder, we can decide whether the
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71 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
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72 * (it will never equal to 1/2ulp).
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73 * The rounding mode can be detected by checking whether
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74 * huge + tiny is equal to huge, and whether huge - tiny is
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75 * equal to huge for some floating point number "huge" and "tiny".
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76 *
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77 * Special cases:
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78 * sqrt(+-0) = +-0 ... exact
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79 * sqrt(inf) = inf
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80 * sqrt(-ve) = NaN ... with invalid signal
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81 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
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82 *
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83 * Other methods : see the appended file at the end of the program below.
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84 *---------------
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85 */
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86
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87 /*#include "math.h"*/
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88 #include "math_private.h"
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89
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90 #ifdef __STDC__
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91 double SDL_NAME(copysign)(double x, double y)
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92 #else
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93 double SDL_NAME(copysign)(x,y)
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94 double x,y;
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95 #endif
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96 {
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97 u_int32_t hx,hy;
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98 GET_HIGH_WORD(hx,x);
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99 GET_HIGH_WORD(hy,y);
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100 SET_HIGH_WORD(x,(hx&0x7fffffff)|(hy&0x80000000));
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101 return x;
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102 }
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103
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104 #ifdef __STDC__
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105 double SDL_NAME(scalbn) (double x, int n)
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106 #else
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107 double SDL_NAME(scalbn) (x,n)
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108 double x; int n;
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109 #endif
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110 {
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111 int32_t k,hx,lx;
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112 EXTRACT_WORDS(hx,lx,x);
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113 k = (hx&0x7ff00000)>>20; /* extract exponent */
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114 if (k==0) { /* 0 or subnormal x */
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115 if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
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116 x *= two54;
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117 GET_HIGH_WORD(hx,x);
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118 k = ((hx&0x7ff00000)>>20) - 54;
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119 if (n< -50000) return tiny*x; /*underflow*/
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120 }
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121 if (k==0x7ff) return x+x; /* NaN or Inf */
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122 k = k+n;
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123 if (k > 0x7fe) return huge*SDL_NAME(copysign)(huge,x); /* overflow */
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124 if (k > 0) /* normal result */
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125 {SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); return x;}
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126 if (k <= -54) {
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127 if (n > 50000) /* in case integer overflow in n+k */
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128 return huge*SDL_NAME(copysign)(huge,x); /*overflow*/
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129 else return tiny*SDL_NAME(copysign)(tiny,x); /*underflow*/
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130 }
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diff changeset
131 k += 54; /* subnormal result */
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132 SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20));
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diff changeset
133 return x*twom54;
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134 }
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diff changeset
135
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diff changeset
136 #ifdef __STDC__
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diff changeset
137 double __ieee754_sqrt(double x)
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diff changeset
138 #else
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diff changeset
139 double __ieee754_sqrt(x)
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diff changeset
140 double x;
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diff changeset
141 #endif
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diff changeset
142 {
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diff changeset
143 double z;
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diff changeset
144 int32_t sign = (int)0x80000000;
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parents:
diff changeset
145 int32_t ix0,s0,q,m,t,i;
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parents:
diff changeset
146 u_int32_t r,t1,s1,ix1,q1;
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diff changeset
147
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diff changeset
148 EXTRACT_WORDS(ix0,ix1,x);
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diff changeset
149
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diff changeset
150 /* take care of Inf and NaN */
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diff changeset
151 if((ix0&0x7ff00000)==0x7ff00000) {
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parents:
diff changeset
152 return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
153 sqrt(-inf)=sNaN */
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diff changeset
154 }
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parents:
diff changeset
155 /* take care of zero */
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parents:
diff changeset
156 if(ix0<=0) {
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parents:
diff changeset
157 if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
158 else if(ix0<0)
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parents:
diff changeset
159 return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
160 }
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diff changeset
161 /* normalize x */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
162 m = (ix0>>20);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
163 if(m==0) { /* subnormal x */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
164 while(ix0==0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
165 m -= 21;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
166 ix0 |= (ix1>>11); ix1 <<= 21;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
167 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
168 for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
169 m -= i-1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
170 ix0 |= (ix1>>(32-i));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
171 ix1 <<= i;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
172 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
173 m -= 1023; /* unbias exponent */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
174 ix0 = (ix0&0x000fffff)|0x00100000;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
175 if(m&1){ /* odd m, double x to make it even */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
176 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
177 ix1 += ix1;
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diff changeset
178 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
179 m >>= 1; /* m = [m/2] */
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diff changeset
180
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diff changeset
181 /* generate sqrt(x) bit by bit */
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diff changeset
182 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
183 ix1 += ix1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
184 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
185 r = 0x00200000; /* r = moving bit from right to left */
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diff changeset
186
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diff changeset
187 while(r!=0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
188 t = s0+r;
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diff changeset
189 if(t<=ix0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
190 s0 = t+r;
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diff changeset
191 ix0 -= t;
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diff changeset
192 q += r;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
193 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
194 ix0 += ix0 + ((ix1&sign)>>31);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
195 ix1 += ix1;
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diff changeset
196 r>>=1;
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diff changeset
197 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
198
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diff changeset
199 r = sign;
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diff changeset
200 while(r!=0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
201 t1 = s1+r;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
202 t = s0;
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diff changeset
203 if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
204 s1 = t1+r;
1345
7f32b9bede06 Fixes for Visual C++ 6.0
Sam Lantinga <slouken@libsdl.org>
parents: 1330
diff changeset
205 if(((int32_t)(t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
1330
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
206 ix0 -= t;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
207 if (ix1 < t1) ix0 -= 1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
208 ix1 -= t1;
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diff changeset
209 q1 += r;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
210 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
211 ix0 += ix0 + ((ix1&sign)>>31);
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diff changeset
212 ix1 += ix1;
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diff changeset
213 r>>=1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
214 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
215
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Sam Lantinga <slouken@libsdl.org>
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diff changeset
216 /* use floating add to find out rounding direction */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
217 if((ix0|ix1)!=0) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
218 z = one-tiny; /* trigger inexact flag */
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Sam Lantinga <slouken@libsdl.org>
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diff changeset
219 if (z>=one) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
220 z = one+tiny;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
221 if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
222 else if (z>one) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
223 if (q1==(u_int32_t)0xfffffffe) q+=1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
224 q1+=2;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
225 } else
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parents:
diff changeset
226 q1 += (q1&1);
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diff changeset
227 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
228 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
229 ix0 = (q>>1)+0x3fe00000;
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diff changeset
230 ix1 = q1>>1;
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diff changeset
231 if ((q&1)==1) ix1 |= sign;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
232 ix0 += (m <<20);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
233 INSERT_WORDS(z,ix0,ix1);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
234 return z;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
235 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
236
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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diff changeset
237 /*
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diff changeset
238 Other methods (use floating-point arithmetic)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
239 -------------
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diff changeset
240 (This is a copy of a drafted paper by Prof W. Kahan
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parents:
diff changeset
241 and K.C. Ng, written in May, 1986)
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diff changeset
242
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243 Two algorithms are given here to implement sqrt(x)
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diff changeset
244 (IEEE double precision arithmetic) in software.
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diff changeset
245 Both supply sqrt(x) correctly rounded. The first algorithm (in
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diff changeset
246 Section A) uses newton iterations and involves four divisions.
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parents:
diff changeset
247 The second one uses reciproot iterations to avoid division, but
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
248 requires more multiplications. Both algorithms need the ability
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
249 to chop results of arithmetic operations instead of round them,
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diff changeset
250 and the INEXACT flag to indicate when an arithmetic operation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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diff changeset
251 is executed exactly with no roundoff error, all part of the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
252 standard (IEEE 754-1985). The ability to perform shift, add,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
253 subtract and logical AND operations upon 32-bit words is needed
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parents:
diff changeset
254 too, though not part of the standard.
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diff changeset
255
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parents:
diff changeset
256 A. sqrt(x) by Newton Iteration
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diff changeset
257
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parents:
diff changeset
258 (1) Initial approximation
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diff changeset
259
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
260 Let x0 and x1 be the leading and the trailing 32-bit words of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
261 a floating point number x (in IEEE double format) respectively
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parents:
diff changeset
262
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
263 1 11 52 ...widths
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
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parents:
diff changeset
264 ------------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
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265 x: |s| e | f |
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266 ------------------------------------------------------
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267 msb lsb msb lsb ...order
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268
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269
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270 ------------------------ ------------------------
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271 x0: |s| e | f1 | x1: | f2 |
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272 ------------------------ ------------------------
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273
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274 By performing shifts and subtracts on x0 and x1 (both regarded
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275 as integers), we obtain an 8-bit approximation of sqrt(x) as
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276 follows.
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277
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278 k := (x0>>1) + 0x1ff80000;
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279 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
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280 Here k is a 32-bit integer and T1[] is an integer array containing
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281 correction terms. Now magically the floating value of y (y's
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282 leading 32-bit word is y0, the value of its trailing word is 0)
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283 approximates sqrt(x) to almost 8-bit.
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284
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285 Value of T1:
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286 static int T1[32]= {
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287 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
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288 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
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289 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
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290 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
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291
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292 (2) Iterative refinement
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293
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294 Apply Heron's rule three times to y, we have y approximates
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295 sqrt(x) to within 1 ulp (Unit in the Last Place):
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296
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297 y := (y+x/y)/2 ... almost 17 sig. bits
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298 y := (y+x/y)/2 ... almost 35 sig. bits
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299 y := y-(y-x/y)/2 ... within 1 ulp
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300
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301
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302 Remark 1.
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303 Another way to improve y to within 1 ulp is:
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304
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305 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
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306 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
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307
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308 2
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309 (x-y )*y
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310 y := y + 2* ---------- ...within 1 ulp
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311 2
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312 3y + x
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313
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314
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315 This formula has one division fewer than the one above; however,
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316 it requires more multiplications and additions. Also x must be
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317 scaled in advance to avoid spurious overflow in evaluating the
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318 expression 3y*y+x. Hence it is not recommended uless division
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319 is slow. If division is very slow, then one should use the
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320 reciproot algorithm given in section B.
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321
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322 (3) Final adjustment
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323
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324 By twiddling y's last bit it is possible to force y to be
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325 correctly rounded according to the prevailing rounding mode
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326 as follows. Let r and i be copies of the rounding mode and
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327 inexact flag before entering the square root program. Also we
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328 use the expression y+-ulp for the next representable floating
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329 numbers (up and down) of y. Note that y+-ulp = either fixed
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330 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
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331 mode.
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332
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333 I := FALSE; ... reset INEXACT flag I
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334 R := RZ; ... set rounding mode to round-toward-zero
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335 z := x/y; ... chopped quotient, possibly inexact
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336 If(not I) then { ... if the quotient is exact
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337 if(z=y) {
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338 I := i; ... restore inexact flag
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339 R := r; ... restore rounded mode
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340 return sqrt(x):=y.
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341 } else {
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342 z := z - ulp; ... special rounding
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343 }
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344 }
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345 i := TRUE; ... sqrt(x) is inexact
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346 If (r=RN) then z=z+ulp ... rounded-to-nearest
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347 If (r=RP) then { ... round-toward-+inf
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348 y = y+ulp; z=z+ulp;
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349 }
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350 y := y+z; ... chopped sum
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351 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
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352 I := i; ... restore inexact flag
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353 R := r; ... restore rounded mode
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354 return sqrt(x):=y.
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355
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356 (4) Special cases
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357
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358 Square root of +inf, +-0, or NaN is itself;
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359 Square root of a negative number is NaN with invalid signal.
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360
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361
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362 B. sqrt(x) by Reciproot Iteration
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363
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364 (1) Initial approximation
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365
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366 Let x0 and x1 be the leading and the trailing 32-bit words of
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367 a floating point number x (in IEEE double format) respectively
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368 (see section A). By performing shifs and subtracts on x0 and y0,
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369 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
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370
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371 k := 0x5fe80000 - (x0>>1);
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372 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
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373
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374 Here k is a 32-bit integer and T2[] is an integer array
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375 containing correction terms. Now magically the floating
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376 value of y (y's leading 32-bit word is y0, the value of
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377 its trailing word y1 is set to zero) approximates 1/sqrt(x)
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378 to almost 7.8-bit.
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379
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380 Value of T2:
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381 static int T2[64]= {
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382 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
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383 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
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384 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
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385 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
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386 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
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387 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
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388 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
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diff changeset
389 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
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390
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391 (2) Iterative refinement
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392
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393 Apply Reciproot iteration three times to y and multiply the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
394 result by x to get an approximation z that matches sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
395 to about 1 ulp. To be exact, we will have
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
396 -1ulp < sqrt(x)-z<1.0625ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
397
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
398 ... set rounding mode to Round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
399 y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
400 y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
401 ... special arrangement for better accuracy
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
402 z := x*y ... 29 bits to sqrt(x), with z*y<1
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
403 z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
404
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
405 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
406 (a) the term z*y in the final iteration is always less than 1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
407 (b) the error in the final result is biased upward so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
408 -1 ulp < sqrt(x) - z < 1.0625 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
409 instead of |sqrt(x)-z|<1.03125ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
410
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
411 (3) Final adjustment
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
412
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
413 By twiddling y's last bit it is possible to force y to be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
414 correctly rounded according to the prevailing rounding mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
415 as follows. Let r and i be copies of the rounding mode and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
416 inexact flag before entering the square root program. Also we
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
417 use the expression y+-ulp for the next representable floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
418 numbers (up and down) of y. Note that y+-ulp = either fixed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
419 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
420 mode.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
421
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
422 R := RZ; ... set rounding mode to round-toward-zero
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
423 switch(r) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
424 case RN: ... round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
425 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
426 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
427 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
428 case RZ:case RM: ... round-to-zero or round-to--inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
429 R:=RP; ... reset rounding mod to round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
430 if(x<z*z ... rounded up) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
431 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
432 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
433 case RP: ... round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
434 if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
435 if(x>z*z ...chopped) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
436 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
437 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
438
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
439 Remark 3. The above comparisons can be done in fixed point. For
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
440 example, to compare x and w=z*z chopped, it suffices to compare
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
441 x1 and w1 (the trailing parts of x and w), regarding them as
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
442 two's complement integers.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
443
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
444 ...Is z an exact square root?
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
445 To determine whether z is an exact square root of x, let z1 be the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
446 trailing part of z, and also let x0 and x1 be the leading and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
447 trailing parts of x.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
448
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
449 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
450 I := 1; ... Raise Inexact flag: z is not exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
451 else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
452 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
453 k := z1 >> 26; ... get z's 25-th and 26-th
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
454 fraction bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
455 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
456 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
457 R:= r ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
458 return sqrt(x):=z.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
459
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
460 If multiplication is cheaper then the foregoing red tape, the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
461 Inexact flag can be evaluated by
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
462
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
463 I := i;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
464 I := (z*z!=x) or I.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
465
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
466 Note that z*z can overwrite I; this value must be sensed if it is
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
467 True.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
468
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
469 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
470 zero.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
471
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
472 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
473 z1: | f2 |
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
474 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
475 bit 31 bit 0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
476
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
477 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
478 or even of logb(x) have the following relations:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
479
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
480 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
481 bit 27,26 of z1 bit 1,0 of x1 logb(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
482 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
483 00 00 odd and even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
484 01 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
485 10 10 odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
486 10 00 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
487 11 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
488 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
489
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
490 (4) Special cases (see (4) of Section A).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
491
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
492 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows,
Sam Lantinga <slouken@libsdl.org>
parents:
diff changeset
493